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Easiest is maybe to just use Poincaré duality, in the form of the statement that $H^k_c(U) \otimes H^{2d-k}(U) \to H^{2d}_c(U) \cong \mathbf Q(-d)$ (where $d = \dim_\mathbf{C} U$) is a perfect pairing, compatible with the mixed Hodge structures. Then $\dim \mathfrak{gr}_F^p\mathfrak{gr}^W_m H^k_c(U)= \dim \mathfrak{gr}_F^{d-p}\mathfrak{gr}^W_{2d-m}H^{2d-k}(U)$, in other words, the Hodge numbers of $H^k_c$ and $H^{2d-k}$ are related by the transformation $(p,q) \leftrightarrow (d-p,d-q)$.